logo

SCIENCE CHINA Technological Sciences, Volume 61 , Issue 3 : 346-358(2018) https://doi.org/10.1007/s11431-016-9070-8

A high order numerical manifold method and its application to linear elastic continuous and fracture problems

More info
  • ReceivedDec 6, 2016
  • AcceptedMay 22, 2017
  • PublishedJul 7, 2017

Abstract

The numerical manifold method (NMM) is a partition of unity (PU) based method. For the purpose of obtaining better accuracy with the same mesh, high order global approximation can be adopted by increasing the order of local approximations (LAs). This, however, will cause the “linear dependence” (LD) issue, where the global matrix is rank deficient even after sufficient constraints are enforced. In this paper, through quadrilateral mesh to form the mathematical cover, a high order numerical manifold method called Quad4-COLS (NMM) is developed, where the constrained and orthonormalized least-squares method (CO-LS) is used to construct the LAs. The developed Quad4-COLS (NMM) does not need extra nodes or DOFs to construct high order global approximations, while is free from the LD issue. Nine numerical tests including five tests for linear elastic continuous problems and four tests for linear elastic fracture problems are carried out to validate the accuracy and robustness of the proposed Quad4-COLS (NMM).


Funded by

This study was supported by the National Natural Science Foundation of China(51609240,11572009)

Zhe Jiang Provincial Natural Science Foundation of China(LY13E080009)

National Basic Research Program of China(2014CB047100)


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51609240 & 11572009), the Zhe Jiang Provincial Natural Science Foundation of China (Grant No. LY13E080009), and the National Basic Research Program of China (Grant No. 2014CB047100).


References

[1] Zienkiewicz OC, Taylor RL. The Finite Element Method. 5th Ed. Oxford: Butterworth-Heinemann, 2000. Google Scholar

[2] Yang Y, Zheng H, Sivaselvan M V. A rigorous and unified mass lumping scheme for higher-order elements. Comp Methods Appl Mech Eng, 2017, 319: 491-514 CrossRef ADS Google Scholar

[3] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Eng Fracture Mech, 2013, 110: 113-137 CrossRef Google Scholar

[4] Areias P, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theor Appl Fracture Mech, 2014, 72: 50-63 CrossRef Google Scholar

[5] Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods. Int J Numer Meth Engng, 1994, 37: 229-256 CrossRef ADS Google Scholar

[6] Zhuang X, Augarde C. Aspects of the use of orthogonal basis functions in the element-free Galerkin method. Int J Numer Meth Engng, 2009, 139 CrossRef Google Scholar

[7] Rabczuk T, Belytschko T, Xiao S P. Stable particle methods based on Lagrangian kernels. Comp Methods Appl Mech Eng, 2004, 193: 1035-1063 CrossRef ADS Google Scholar

[8] Zhuang X, Cai Y, Augarde C. A meshless sub-region radial point interpolation method for accurate calculation of crack tip fields. Theor Appl Fracture Mech, 2014, 69: 118-125 CrossRef Google Scholar

[9] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. Int J Numer Meth Engng, 2004, 61: 2316-2343 CrossRef ADS Google Scholar

[10] Bordas S, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Eng Fracture Mech, 2008, 75: 943-960 CrossRef Google Scholar

[11] Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: Framework and modeling. Int J Numer Meth Engng, 2012, 92: 969-998 CrossRef ADS Google Scholar

[12] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Meth Engng, 1999, 45: 601-620 CrossRef Google Scholar

[13] Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Meth Engng, 1999, 46: 131-150 CrossRef Google Scholar

[14] Strouboulis T, Babuška I, Copps K. The design and analysis of the generalized finite element method. Comp Methods Appl Mech Eng, 2000, 181: 43-69 CrossRef Google Scholar

[15] Babuška I, Melenk J M. The partition of unity method. Int J Numer Meth Engng, 1997, 40: 727-758 CrossRef Google Scholar

[16] Talebi H, Samaniego C, Samaniego E, et al. On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. Int J Numer Meth Engng, 2012, 89: 1009-1027 CrossRef ADS Google Scholar

[17] Sukumar N, Chopp D L, Moës N, et al. Modeling holes and inclusions by level sets in the extended finite-element method. Comp Methods Appl Mech Eng, 2001, 190: 6183-6200 CrossRef ADS Google Scholar

[18] Sukumar N, Moës N, Moran B, et al. Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Engng, 2000, 48: 1549-1570 CrossRef Google Scholar

[19] Elguedj T, Gravouil A, Maigre H. An explicit dynamics extended finite element method. Part 1: Mass lumping for arbitrary enrichment functions. Comp Methods Appl Mech Eng, 2009, 198: 2297-2317 CrossRef ADS Google Scholar

[20] Menouillard T, Réthoré J, Moës N, et al. Mass lumping strategies for X-FEM explicit dynamics: Application to crack propagation. Int J Numer Meth Engng, 2008, 74: 447-474 CrossRef ADS Google Scholar

[21] Ghorashi S S, Valizadeh N, Mohammadi S, et al. T-spline based XIGA for fracture analysis of orthotropic media. Comp Struct, 2015, 147: 138-146 CrossRef Google Scholar

[22] Bordas S P A, Rabczuk T, Hung N X, et al. Strain smoothing in FEM and XFEM. Comp Struct, 2010, 88: 1419-1443 CrossRef Google Scholar

[23] Fries T P, Belytschko T. The extended/generalized finite element method: An overview of the method and its applications. Int J Numer Meth Engng, 2010, 1‒3 CrossRef Google Scholar

[24] Duarte C A, Hamzeh O N, Liszka T J, et al. A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comp Methods Appl Mech Eng, 2001, 190: 2227-2262 CrossRef ADS Google Scholar

[25] Shi GH. Manifold method of material analysis. In: Proceedings of the Transcations of the Ninth Army Confernece on Applied Mathematics and Computing. Minneapolis, 1991. 57–76. Google Scholar

[26] Zheng H, Liu Z, Ge X. Numerical manifold space of Hermitian form and application to Kirchhoff’s thin plate problems. Int J Numer Meth Engng, 2013, 95: 721-739 CrossRef ADS Google Scholar

[27] Zheng H, Liu F, Li C. Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method. Appl Math Model, 2015, 39: 794-808 CrossRef Google Scholar

[28] Fan L F, Yi X W, Ma G W. Numerical manifold method (NMM) simulation of stress wave propagation through fractured rock mass. Int J Appl Mech, 2013, 05: 1350022 CrossRef ADS Google Scholar

[29] Yang Y T, Zheng H. Direct approach to treatment of contact in numerical manifold method. Int J Geomechan, 2016: E4016012. Google Scholar

[30] Zheng H, Yang Y. On generation of lumped mass matrices in partition of unity based methods. Int J Numer Meth Engng, 2017, 198 CrossRef Google Scholar

[31] Zheng H, Liu F, Du X. Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method. Comp Methods Appl Mech Eng, 2015, 295: 150-171 CrossRef ADS Google Scholar

[32] Zhang H H, Li L X, An X M, et al. Numerical analysis of 2-D crack propagation problems using the numerical manifold method. Eng Anal Boundary Elements, 2010, 34: 41-50 CrossRef Google Scholar

[33] Yang Y, Tang X, Zheng H, et al. Three-dimensional fracture propagation with numerical manifold method. Eng Anal Bound Elem, 2016, 72: 65-77 CrossRef Google Scholar

[34] Fan H, He S, Jiang Z. A high-order numerical manifold method with nine-node triangular meshes. Eng Anal Bound Elem, 2015, 61: 172-182 CrossRef Google Scholar

[35] Tian R. Extra-dof-free and linearly independent enrichments in GFEM. Comp Methods Appl Mech Eng, 2013, 266: 1-22 CrossRef Google Scholar

[36] Tian R, Yagawa G, Terasaka H. Linear dependence problems of partition of unity-based generalized FEMs. Comp Methods Appl Mech Eng, 2006, 195: 4768-4782 CrossRef ADS Google Scholar

[37] Zhang G X, Sugiura Y, Hasegawa H, et al. The second order manifold method with six node triangle mesh. Struct Eng/Earthq Eng, 2002, 19: 1s-9s CrossRef Google Scholar

[38] Zheng H, Xu D. New strategies for some issues of numerical manifold method in simulation of crack propagation. Int J Numer Meth Engng, 2014, 97: 986-1010 CrossRef ADS Google Scholar

[39] Yang Y, Zheng H. A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis. Eng Fracture Mech, 2016, 162: 51-75 CrossRef Google Scholar

[40] Xu J P, Rajendran S. A ‘FE-Meshfree’ TRIA3 element based on partition of unity for linear and geometry nonlinear analyses. Comput Mech, 2013, 51: 843-864 CrossRef ADS Google Scholar

[41] Tang X, Zheng C, Wu S, et al. A novel four-node quadrilateral element with continuous nodal stress. Appl Math Mech-Engl Ed, 2009, 30: 1519-1532 CrossRef Google Scholar

[42] Rajendran S, Zhang B R. A “FE-meshfree” QUAD4 element based on partition of unity. Comp Methods Appl Mech Eng, 2007, 197: 128-147 CrossRef ADS Google Scholar

[43] Yang Y, Tang X, Zheng H. Construct ‘FE-Meshfree’ Quad4 using mean value coordinates. Eng Anal Bound Elem, 2015, 59: 78-88 CrossRef Google Scholar

[44] Yang Y, Xu D, Zheng H. A partition-of-unity based ‘FE-Meshfree’ triangular element with radial-polynomial basis functions for static and free vibration analysis. Eng Anal Bound Elem, 2016, 65: 18-38 CrossRef Google Scholar

[45] Nguyen N T, Bui T Q, Zhang C, et al. Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method. Eng Anal Bound Elem, 2014, 44: 87-97 CrossRef Google Scholar

[46] Erdogan F, Sih G C. On the crack extension in plates under plane loading and transverse shear. J Basic Eng, 1963, 85: 519-527 CrossRef Google Scholar

[47] Timoshenko S P, Goodier J N. Theory of Elasticity. 3rd Ed. New York, U.K.: Mcgraw-Hill College, 1970. Google Scholar

[48] Xu J P, Rajendran S. A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses. Comp Methods Appl Mech Eng, 2011, 200: 3309-3323 CrossRef ADS Google Scholar

[49] Liu GR, Nguyen-Thoi T. Smoothed Finite Element Methods. New York: CRC Press, 2010. Google Scholar

[50] Yang Y, Bi R, Zheng H. A hybrid ‘FE-Meshless’ QUAD4 with continuous nodal stress using radial-polynomial basis functions. Eng Anal Bound Elem, 2015, 53: 73-85 CrossRef Google Scholar

[51] Ewalds H, Wanhill R. Fracture Mechanics. New York: Edward Arnold, 1989. Google Scholar

[52] Kang Z, Bui T Q, Nguyen D D, et al. An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics. Acta Mech, 2015, 226: 3991-4015 CrossRef Google Scholar

[53] Tang X, Wu S, Zheng C, et al. A novel virtual node method for polygonal elements. Appl Math Mech-Engl Ed, 2009, 30: 1233-1246 CrossRef Google Scholar

[54] Leonel E D, Venturini W S. Multiple random crack propagation using a boundary element formulation. Eng Fracture Mech, 2011, 78: 1077-1090 CrossRef Google Scholar

  • Figure 1

    (Color online) Mathematical mesh, mathematical patches, physical patches and manifold elements.

  • Figure 2

    (Color online) Definition of support domain for a physical patch to obtain its support nodes.

  • Figure 3

    (Color online) Meshes used for checking linear dependence issue. △: Constrains in both the x- and y-directions; ○: constrains in the y-direction.

  • Figure 4

    A cantilever beam subjected to a tip-shear force on the right end.

  • Figure 5

    (Color online) Mesh for cantilever beam subjected to a tip-shear force. (a) Mesh A (175 nodes); (b) Mesh B (297 nodes); (c) Mesh C (451 nodes).

  • Figure 6

    (Color online) Comparison of accuracy for cantilever beam problem subjected to a tip-shear force. (a) Relative error in displacement norm; (b) relative error in energy norm.

  • Figure 7

    A 2D cantilever beam subjected to a tip-moment.

  • Figure 8

    (Color online) Comparison of accuracy for cantilever beam problem subjected to a tip-moment. (a) Relative error in displacement norm; (b) relative error in energy norm.

  • Figure 9

    (Color online) Cook’s skew beam. (a) The geometry and the boundary conditions; (b) example mesh for Trig3 (NMM) with 4×4 layers and 22 nodes; (c) example mesh for ES-FEM and NS-FEM with 4×4 layers and 25 nodes; (d) example mesh for Quad8 with 4×4 layers and 65 nodes; (e) example mesh for Quad4 (NMM) and Quad4-COLS2 (NMM) with 4×4 layers and 22 nodes.

  • Figure 10

    (Color online) Displacement of Cook’s skew beam.

  • Figure 11

    (Color online) Relative displacement error of Cook’s skew beam.

  • Figure 12

    Dimensions of slope model [50].

  • Figure 13

    (Color online) Discretized model of a slope. (a) 166 PPs; (b) 597 PPs; (c) 2257 PPs.

  • Figure 14

    (Color online) Displacement errors of the slope. (a) Horizontal displacement errors of point B; (b) vertical displacement errors of point A.

  • Figure 15

    (Color online) An edge-cracked plate subjected to a uniform tensile loading. (a) The geometry and the boundary conditions; (b) example mesh with 19×39 layers.

  • Figure 16

    (Color online) A three-point bending test specimen. (a) The geometry and the boundary conditions; (b) a mesh with 81×16 layers.

  • Figure 17

    Propagation of the fracture path for a three-point bending test specimen.

  • Figure 18

    (Color online) A beam under four-point loading. (a) The geometry and the boundary conditions; (b) a mesh with 80×20 layers.

  • Figure 19

    Propagation of the fracture path in a beam under four-point loading by the Quad4-COLS2 (NMM). (a) Step 1; (b) step 3; (c) step 6; (d) step 9.

  • Figure 20

    (Color online) A perforated plate with a circle hole subjected to a uniform tensile loading.

  • Figure 21

    (Color online) Mesh for plate with a circle hole subjected to a uniform tensile loading (Mesh with 25×76 layers,1962 PPs).

  • Figure 22

    Propagation of the fracture path in a perforated panel subjected to a uniform tensile loading by the Quad4-COLS2 (NMM). (a) Step 0; (b) step 1; (c) step 3; (d) step 5; (e) step 7; (f) step 9.

  • Table 1   Comparison of rank deficiency for Quad4-COLS1 (NMM) and Quad4-COLS2 (NMM)

    Mesh a)

    Total DOfs

    Quad4-COLS1 (NMM)

    Quad4-COLS2 (NMM)

    Before constraints

    After constraints

    Before constraints

    After constraints

    (a)

    8

    3

    0

    3

    0

    (b)

    12

    3

    0

    3

    0

    (c)

    18

    3

    0

    3

    0

    (d)

    24

    3

    0

    3

    0

    (e)

    32

    3

    0

    3

    0

    The meshes are corresponding to Figure 3.

  • Table 2   Vertical displacement for point for Cook’s skew beam

    Element type

    Mesh

    4×4

    8×8

    16×16

    32×32

    Trig3 (NMM)

    n

    22

    60

    183

    622

    V A

    16.93

    21.20

    23.07

    23.70

    Quad4 (NMM)

    n

    22

    60

    183

    622

    V A

    19.72

    22.56

    23.54

    23.85

    ES-FEM

    n

    25

    81

    289

    1089

    V A

    19.72

    22.74

    23.65

    23.88

    NS-FEM

    n

    25

    81

    289

    1089

    V A

    26.41

    24.85

    24.24

    24.05

    Quad8

    n

    65

    225

    833

    3201

    V A

    23.71

    23.88

    23.93

    23.96

    Quad4-COLS2 (NMM)

    n

    22

    60

    183

    622

    V A

    22.94

    23.62

    23.82

    23.90

    Reference [48]

    23.96

  • Table 3   Convergence of normalized SIFs for an edge crack under tensile loading

    Element type

    Normalized SIFs

    Mesh

    19×39

    29×59

    39×79

    Quad4-COLS2 (NMM)

    M I

    0.973

    0.989

    0.996

    XQ4 [52]

    M I

    0.972

    0.983

    0.989

    XCQ4 [52]

    M I

    0.976

    0.988

    0.993

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备17057255号       京公网安备11010102003388号